Spin and spin-wave dynamics in Josephson junctions

We extend the Keldysh formulation to quantum spin systems and derive exact equations of motion. This allows us to explore the dynamics of single spins and of ferromagnets when these are inserted between superconducting leads. Several new effects are reported. Chief amongst these are nutations of single $S=1∕2$ spins in Josephson junctions. These nutations are triggered by the superconducting pairing correlations in the leads. Similarly, we find that on rather universal grounds, magnets display unconventional spin wave dynamics when placed in Josephson junctions. These lead to modifications in the tunneling current.

Figure 1

The standard Keldysh contour. The times $T$ and $T^{\prime }$ are taken to be $-\infty$ and $\infty$, respectively. The form of this contour will be heavily employed in our work when time ordering various spin products.

Figure 2

The magnetic spin coupled to two superconducting leads.

Figure 3

The resulting spin motion on the unit sphere in the general case. As in the motion of classical spinning top, the spin exhibits undulations along the polar direction. As a consequence of entanglement with the tunneling electrons, the magnitude of the spin is not constant—the spin further “breathes” in and out as it nutates.

Figure 4

A ferromagnetic slab inserted between two superconducting leads. The entire system is subjected to a weak external magnetic field $B$. A schematic of the precessing spins is shown.

Figure 5

A SQUID-based detection scheme. The SQUID monitors the magnetic field produced by the magnetic cluster in one of the junctions.